Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $\xi_{m,k}$ of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed m estimates on the area $\xi_{m,k}$ in terms of their genus $g = mk \gg 1$.

Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $\xi_{m,k}$ of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed $m$ estimates on the area $\xi_{m,k}$ in terms of their genus $g = mk \gg 1$.

We consider unbranched Willmore surfaces in Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index—the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three‑space. This Index is $m-d$, where $m$ is the number of ends of the corresponding complete minimal surface and $d$ is the dimension of the span of the normals at the $m$-fold point. The dimension $d$ is either two or three. For $m=4$, we prove that $d=3$. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.

In this article we prove the strong Morse inequalities for the area functional in codimension one, assuming that the ambient dimension satisfies $3 \leq (n + 1) \leq 7$, in both the closed and the boundary cases.

We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity $\mathbb{C}^n / G$ for a finite subgroup $G \subset SL(n, \mathbb{C})$ and any crepant resolution $Y$, we prove that the rank of positive symplectic cohomology $SH^{\ast}_{+} (Y)$ is the number $\lvert \operatorname{Conj}(G) \rvert$ of conjugacy classes of $G$, and that twice the age grading on conjugacy classes is the $\mathbb{Z}$-grading on $SH^{\ast-1}_{+} (Y)$ by the Conley–Zehnder index. The generalized McKay correspondence follows as $SH^{\ast-1}_{+} (Y)$ is naturally isomorphic to ordinary cohomology $H^\ast (Y)$, due to a vanishing result for full symplectic cohomogy. In the Appendix we construct a novel filtration on the symplectic chain complex for any non-exact convex symplectic manifold, which yields both a Morse–Bott spectral sequence and a construction of positive symplectic cohomology.

In this paper we prove that the generic singularities of mean curvature flow of closed embedded surfaces in $\mathbb{R}^3$ modelled by closed self-shrinkers with multiplicity has multiplicity one. Together with the previous result by Colding–Minicozzi in [CM12], we conclude that the only generic singularity of mean curvature flow of closed embedded surfaces in $\mathbb{R}^3$ modelled by closed self-shrinkers is a multiplicity one sphere. We also construct particular perturbations of the flow to avoid those singularities with multiplicity higher than one. Our result partially addresses the well-known multiplicity one conjecture by Ilmanen.